121:, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a
495:
177:
586:
is a pseudocomplemented (and distributive) lattice with the meet and join being the usual union and intersection of open sets. The pseudocomplement of an open set
762:
726:
663:. In general, an implicative lattice may not have a minimal element. If such a minimal element exists, then each pseudocomplement
435:
330:
721:. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119.
680:
834:
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is pseudocomplemented. In fact, a Stone algebra can be defined as a pseudocomplemented distributive lattice
448:
412:
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350:
40:
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465:
147:
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575:
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44:
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298:
51:
828:
784:
455:
32:
796:
128:. However this latter term may have other meanings in other areas of mathematics.
17:
779:
118:
28:
342:
603:
579:
191:
117:
is pseudocomplemented. Every pseudocomplemented lattice is necessarily
655:. A lattice with the pseudocomplement for each two elements is called
602:. Furthermore, the dense elements of this lattice are exactly the
462:
in which any of the following equivalent statements hold for all
667:* could be defined using relative pseudocomplement as
468:
150:
755:
Universal
Algebra: Fundamentals and Selected Topics
333:(the complement in this algebra is *). In general,
489:
171:
438:; indeed, so do pseudocomplemented semilattices.
8:
748:
746:
744:
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719:Lattices and Ordered Algebraic Structures
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149:
407:), the set of all the dense elements in
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39:is one generalization of the notion of
7:
782:(September 1970). "Stone Lattices".
434:Pseudocomplemented lattices form a
423:-algebra is Boolean if and only if
194:. In particular, 0* = 1 and 1* = 0.
25:
817:(3rd ed.). AMS. p. 44.
1:
797:10.1215/S0012-7094-70-03768-3
757:. CRC Press. pp. 63–70.
391:. Every element of the form
851:
681:Topological vector lattice
111:pseudocomplemented lattice
753:Clifford Bergman (2011).
617:relative pseudocomplement
611:Relative pseudocomplement
606:in the topological sense.
490:{\displaystyle x,y\in L:}
172:{\displaystyle x,y\in L:}
383:* = 0 (or equivalently,
77:with the property that
567:is pseudocomplemented.
491:
451:is pseudocomplemented.
173:
85:* = 0. More formally,
627:is a maximal element
506:) is a sublattice of
492:
174:
466:
449:distributive lattice
148:
113:if every element of
717:T.S. Blyth (2006).
657:implicative lattice
109:itself is called a
105:= 0 }. The lattice
661:Brouwerian lattice
604:dense open subsets
487:
387:** = 1) is called
379:with the property
305:and together with
169:
66:if there exists a
62:is said to have a
31:, particularly in
18:Brouwerian lattice
811:Birkhoff, Garrett
778:Balbes, Raymond;
764:978-1-4398-5129-6
728:978-1-84628-127-3
578:, the (open set)
576:topological space
419:. A distributive
16:(Redirected from
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364:) is the set of
281:} is called the
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68:greatest element
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791:(3): 537–545.
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399:* is dense.
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368:elements of
366:complemented
361:
357:
353:
351:distributive
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329:*)* forms a
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33:order theory
26:
647:is denoted
341:) is not a
29:mathematics
687:References
631:such that
356:-algebra,
343:sublattice
144:, for all
132:Properties
41:complement
479:∈
431:) = {1}.
297:) is a ∧-
161:∈
140:-algebra
89:* = max{
829:Category
813:(1973).
675:See also
592:interior
580:topology
442:Examples
283:skeleton
261:The set
205:** is a
197:The map
192:antitone
182:The map
126:-algebra
643:. This
594:of the
590:is the
558:** = 1.
436:variety
349:. In a
321:)** = (
207:closure
119:bounded
45:lattice
43:. In a
761:
725:
563:Every
540:)** =
454:Every
413:filter
269:) ≝ {
249:)** =
671:→ 0.
659:, or
574:is a
544:** ∨
521:)* =
411:is a
389:dense
273:** |
253:** ∧
230:)* =
190:* is
136:In a
50:with
759:ISBN
723:ISBN
554:* ∨
525:* ∨
325:* ∧
234:* ∧
219:***.
215:* =
73:* ∈
35:, a
793:doi
619:of
598:of
582:on
570:If
548:**;
415:of
345:of
313:= (
301:of
289:.
285:of
257:**.
27:In
831::
789:37
737:^
695:^
615:A
529:*;
395:∨
372:.
309:∪
277:∈
238:*.
201:↦
186:↦
101:∧
97:|
93:∈
81:∧
58:∈
799:.
795::
767:.
731:.
669:a
665:a
653:b
651:→
649:a
641:b
639:≤
637:c
635:∧
633:a
629:c
625:b
621:a
600:A
588:A
584:X
572:X
556:x
552:x
546:y
542:x
538:y
536:∨
534:x
532:(
527:y
523:x
519:y
517:∧
515:x
513:(
510:;
508:L
504:L
502:(
500:S
485::
482:L
476:y
473:,
470:x
460:L
429:L
427:(
425:D
421:p
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403:(
401:D
397:x
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385:x
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370:L
362:L
360:(
358:S
354:p
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337:(
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327:y
323:x
319:y
317:∨
315:x
311:y
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293:(
291:S
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275:x
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265:(
263:S
255:y
251:x
247:y
245:∧
243:x
241:(
236:y
232:x
228:y
226:∨
224:x
222:(
217:x
213:x
209:.
203:x
199:x
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184:x
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